 (199d) Exact Solutions of Population Balances Equations Via An Auxiliary Equation Method for Growth, Nucleation, Breakage and Aggregation Processes | AIChE # (199d) Exact Solutions of Population Balances Equations Via An Auxiliary Equation Method for Growth, Nucleation, Breakage and Aggregation Processes

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## Authors

Groep T – Leuven Engineering College (Associatie KU Leuven)
Loughborough University
Groep T – Leuven Engineering College (Associatie KU Leuven)

There has been a growing interest in the field of population balance equations (PBEs), especially for applications in processes such as granulation and crystallization (Immanuel and Doyle III, 2005; Ma et al., 2002), in aerosol science and biochemical research (Kim and Seinfeld, 1992; Roussos and Kiparissides, 2012). A continuous population balance equation (PBE) involving particulate growth and aggregation, growth, aggregation and breakage, aggregation, nucleation, breakage and growth are investigated in this study. Usually numerical methods are employed to solve these PBEs, because of the absence of an analytical solution. However,  to overcome this, a new technique based on auxiliary equation method (AEM) with a sixth-degree (Pınar and Öziş, 2013a,b) even-power polynomial is proposed to obtain the exact solutions of PBEs under various conditions of growth, nucleation, aggregation and breakage. The proposed method is already applied to growth, nucleation and aggregation processes by Dutta et al. (2013). The proposed method has been found superior to standard numerical methods as it provides analytical solutions for a wide range of initial and boundary conditions. To illustrate the validity and efficiency of the proposed method, analytical solutions obtained with AEM are compared with the numerical solutions, and with the analytical solution as found in the standard literature.

References:

C.D. Immanuel, F.J. Doyle (2005). Solution technique for a multidimensional population balance model  describing granulation processes. Powder Technology, 156. 213–225.
D.L. Ma, D.K. Tafti, R.D. Braatz (2002). High resolution simulation of multi-dimensional crystallization. Ind. Eng. Chem. Res. 41, 6217-6223.
Y.P. Kim, J.H. Seinfeld (1992). Simulation of multicomponent aerosol dynamics. J. Colloid Interface Science 149, 425-449.
A.I. Roussos, C. Kiparissides (2012). A bivariate population balance model for the microbial production of poly(3-hydroxybutyrate). Chemical Engineering Science, 70, 45–53.
Z.Pınar, T.Öziş, (2013a) An  Observation on the Periodic Solutions to Nonlinear Physical models by means of the auxiliary equation with a sixth-degree nonlinear term, Commun Nonlinear Sci Numer Simulat18 2177–2187.
Z.Pınar, T.Öziş, (2013b) The Periodic Solutions to Kawahara Equation by means of the auxiliary equation with a sixth-degree nonlinear term, Journal of Mathematics (http://dx.doi.org/10.1155/2013/106349).
A. Dutta, Z. Pinar, Z., B. Guid., T. Öziş (2013). An auxiliary equation approach to exact solutions of population balance equations for growth, nucleation and aggregation processes. International Conference on Applied Analysis and Mathematical Modelling (ICAAMM). Istanbul, 2-5 June 2013.